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This article gives pointers to special cases of Stone duality and explains a very general instance thereof in detail. OVERVIEW OF STONE-TYPE DUALITIES Probably the most general duality which is classically referred to as "Stone duality" is the duality between the category Sob of Sober Space s with Continuous Function s and the category '''SFrm''' of spatial Frames with appropriate frame homomorphisms. The Dual Category of '''SFrm''' is the category of Locales denoted by '''SLoc'''. The Categorical Equivalence of Sob and '''SLoc''' is the basis for the mathematical area of Pointless Topology , that is devoted to the study of '''Loc''' - the category of all locales of which '''SLoc''' is a full subcategory. The involved constructions are characteristic for this kind of dualities and are therefore detailed below. Now one can easily obtain a number of other dualities by restricting to certain special classes of sober spaces:
Many other Stone-type dualities could be added to these basic dualities. DUALITY OF SOBER SPACES AND SPATIAL LOCALES This section motivates and explains one of the most basic constructions of Stone duality: the duality between topological spaces which are ''sober'' and frames (i.e. Complete Heyting Algebra s) which are ''spatial''. This classical piece of mathematics requires a substantial amount of abstraction that usually tends to puzzle beginners. It should therefore be considered as graduate level mathematics. Some prior exposure to the basics of Category Theory is recommended, although a deep understanding of the concepts of adjunction and duality may well arise from examples such as the result below. Furthermore, concepts of Topology and Order Theory are naturally involved as well, where the later is probably more important for a thorough understanding. The lattice of open sets The starting point for the theory is the fact that every topological space is characterized by a set of points ''X'' and a system Ω(''X'') of within which Suprema and Infima of finite sets are given by set unions and finite set intersections, respectively. Furthermore, it contains both ''X'' and the Empty Set . Since the Embedding of Ω(''X'') into the powerset lattice of ''X'' Preserves finite infima and arbitrary suprema, Ω(''X'') inherits the following distributivity law: : for every element (open set) ''x'' and every subset ''S'' of Ω(''X''). Hence Ω(''X'') is not an arbitrary complete lattice but a ''complete Heyting algebra'' (also called ''frame'' or ''locale'' - the various names are primarily used to distinguish several categories that have the same class of objects but different morphisms: frame morphisms, locale morphisms and homomorphisms of complete Heyting algebras). Now an obvious question is: To what extent is a topological space characterized by its locale of open sets? As already hinted at above, one can go even further. The category Top of topological spaces has as morphisms the continuous functions, where a function ''f'' is continuous if the Inverse Image ''f'' −1(''O'') of any open set in the Codomain of ''f'' is open in the Domain of ''f''. Thus any continuous function ''f'' from a space ''X'' to a space ''Y'' defines an inverse mapping ''f'' −1 from Ω(''Y'') to Ω(''X''). Furthermore, it is easy to check that ''f'' −1 (like any inverse image map) preserves finite intersections and arbitrary unions and therefore is a ''morphism of frames''. If we define Ω(''f'') = ''f'' −1 then Ω becomes a Contravariant Functor from the category Top to the category '''Frm''' of frames and frame morphisms. Using the tools of category theory, the task of finding a characterization of topological spaces in terms of their open set lattices is equivalent to finding a functor from '''Frm''' to Top which is Adjoint to Ω. Points of a locale The goal of this section is to define a functor pt from Frm to '''Top''' that in a certain sense "inverts" the operation of Ω by assigning to each locale ''L'' a set of points pt(''L'') (hence the notation pt) with a suitable topology. But how can we recover the set of points just from the locale, though it is not given as a lattice of sets? It is certain that one cannot expect in general that pt can reproduce all of the original elements of a topological space just from its lattice of open sets - for example all sets with the Indiscrete Topology yield (up to isomorphism) the same locale, such that the information on the specific set is no longer present. However, there is still a reasonable technique for obtaining "points" from a locale, which indeed gives an example of a central construction for Stone-type duality theorems. Let us first look at the points of a topological space ''X''. One is usually tempted to consider a point of ''X'' as an element ''x'' of the set ''X'', but there is in fact a more useful description for our current investigation. Any point ''x'' gives rise to a continuous function ''p''''x'' from the one element topological space 1 (all subsets of which are open) to the space ''X'' by defining ''p''''x''(1) = ''x''. Conversely, any function from 1 to ''X'' clearly determines one point: the element that it "points" to. Therefore the set of points of a topological space is equivalently characterized as the set of functions from 1 to ''X''. When using the functor Ω to pass from Top to '''Frm''', all set-theoretic elements of a space are lost, but - using a fundamental idea of category theory - one can as well work on the Function Space s. Indeed, any "point" ''p''''x'': 1 → ''X'' in Top is mapped to a morphism Ω(''p''''x''): Ω(''X'') → Ω(''1''). The open set lattice of the one-element topological space Ω(''1'') is just (isomorphic to) the two-element locale 2 = { 0, 1 } with 0 < 1. After these observations it appears reasonable to define the set of points of a locale ''L'' to be the set of frame morphisms from ''L'' to 2. Yet, there is no guarantee that every point of the locale Ω(''X'') is in one-to-one correspondence to a point of the topological space ''X'' (consider again the indiscrete topology, for which the open set lattice has only one "point"). Before defining the required topology on pt(''X''), it is worthwhile to clarify the concept of a point of a locale further. The perspective motivated above suggests to consider a point of a locale ''L'' as a frame morphism ''p'' from ''L'' to 2. But these morphisms are characterized equivalently by the inverse images of the two elements of 2. From the properties of frame morphisms, one can derive that ''p'' −1(0) is a lower set (since ''p'' is Monotone ), which contains a greatest element ''a''''p'' = V ''p'' −1(0) (since ''p'' preserves arbitrary suprema). In addition, the Principal Ideal ''p'' −1(0) is a Prime Ideal since ''p'' preserves finite infima and thus the principal ''a''''p'' is a Meet-prime Element . Now the set-inverse of ''p'' −1(0) given by ''p'' −1(1) is a Completely Prime Filter because ''p'' −1(0) is a principal prime ideal. It turns out that all of these descriptions uniquely determine the initial frame morphism. We sum up: A point of a locale ''L'' is equivalently described as:
All of these descriptions have their place within the theory and it is convenient to switch between them as needed. The functor pt Now that a set of points is available for any locale, it remains to equip this set with an appropriate topology in order to define the object part of the functor pt. This is done by defining the open sets of pt(''L'') as | ||
| For Every Element ''a'' Of ''L'' Here We Viewed The Points Of ''L'' As Morphisms, But One Can Of Course State A Similar Definition For All Of The Other Equivalent Characterizations It Can Be Shown That Setting &Omega(pt(''L'')) | {&phi(''a'') ''a'' in ''L''} does really yield a topological space (pt(''L''), &Omega(pt(''L''))) It is common to abbreviate this space as pt(''L'') |
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